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On the topological product of discrete \(\lambda\)-compact spaces. (English) Zbl 0114.14102
General Topology and its Relations to modern Analysis and Algebra, Proc. Sympos. Prague 1961, 148-151 (1962).
[For the entire collection see Zbl 0111.35001.]
A topological space \({\mathfrak X}\) is said to be \(\kappa\)-compact if every class \({\mathcal M}\) of closed subsets of \({\mathfrak X}\) with void intersection contains a subclass \({\mathcal M}' \subseteq {\mathcal M}\) having a void intersection and a power \(\overline{\overline{\mathcal M'}}\) with \(\overline {\overline{\mathcal M'}}< \aleph_\kappa\).
For each cardinal number \(m\) and each pair of ordinal numbers \(\lambda,\kappa\), one uses the abbreviation \(\top(m,\lambda) \to \kappa\) of the statment ”if \({\mathcal F}\) is a class of discrete \(\lambda\)-compact topological spaces with the power \(\overline{\overline{\mathcal F}}=m\) then the topological product of the elements of \({\mathcal F}\) is \(\kappa\)-compact”. The authors give an outline of the proof of the theorem ”if \(\alpha, \gamma\) are ordinals such that \(\aleph_{\alpha+\gamma}\) is singular and \(cf(\gamma) < \omega\) then the statement \(\top(\aleph_{\alpha+\gamma},\alpha +1) \to \alpha+\gamma\) is false” (using the generalized continuum-hypothesis); they discuss some related problems, too. By the theorem the question ”\(\top (\aleph_{\omega},1) \to \omega\)?”, stated in another paper of the authors [see Acta Math. Acad. Sci. Hung. 12, 87-123 (1961; Zbl 0201.32801)], is answered negative.
Reviewer: G.Grimeisen

54D45 Local compactness, \(\sigma\)-compactness
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)